91Ó°ÊÓ

In simple harmonic motion, an object can exhibit circular motion due to its position vector. When dealing with the position vector \(\vec{r} = A \sin \omega t \hat{\imath} + A \cos \omega t \hat{\jmath}\), we see that it is composed of sine and cosine functions. These trigonometric functions represent the x and y components of motion in a plane.

When you compute the magnitude of this position vector, it's akin to finding the hypotenuse of a right triangle with sides \(A \sin \omega t\) and \(A \cos \omega t\). This results in:

• \(\vec{r} \cdot \vec{r} = A^2 \sin^2 \omega t + A^2 \cos^2 \omega t\)

By applying the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), the magnitude simplifies to \(|\vec{r}| = A\). This shows that the object traces a circle with radius \(A\), keeping the distance from the origin constant as it moves.

This fixed distance is what we recognize as uniform circular motion—a key feature of simple harmonic motion.

The velocity of an object in motion is a vector quantity that combines its speed and direction of travel. For this harmonic motion scenario, the velocity\(\vec{v}\) is found by differentiating the position vector with respect to time.

By taking the derivative, the velocity vector becomes:

• \(\vec{v} = \frac{d \vec{r}}{dt} = A \omega \cos \omega t \hat{\imath} - A \omega \sin \omega t \hat{\jmath} \)

This indicates that the velocity acts tangentially to the circular path and changes direction as the object moves around the circle.

The speed, however, is the magnitude of this velocity vector, and it can be calculated as:

• \(\vec{v} \cdot \vec{v} = A^2 \omega^2 \cos^2 \omega t + A^2 \omega^2 \sin^2 \omega t\)

Using the identity \(\sin^2 x + \cos^2 x = 1\), simplifies it to \(|\vec{v}| = A\omega\). This highlights that the speed of the object remains constant, irrespective of the specific position on its circular path.

Angular speed refers to how fast the angle is changing as an object moves along a circular path. In this context, the angular speed is a constant value due to the simplicity of the harmonic motion.

• The previously calculated speed for the object was \(A\omega\) and its circular path radius was \(A\).

The angular speed \(\omega_{angular}\) is determined by dividing the linear speed by the radius:

• \(\omega_{angular} = \frac{A\omega}{A}= \omega\)

This expression reveals that the angular speed in harmonic motion remains equivalent to the original angular frequency of the motion \(\omega\).

This constancy signifies uniform angular velocity, showcasing that the object consistently rotates through its circle at the same rate, reflecting another hallmark of simple harmonic motion.

Mathematical derivation in the context of simple harmonic motion involves using trigonometric identities and derivatives to arrive at equations that explain physical phenomena. Here, we've seen this process in several key aspects.

Initially, the use of the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) was crucial for proving the circular path of the object's motion by confirming the fixed radius \(A\).

Next, by applying calculus through differentiation, we found the velocity expression from the position vector, demonstrating how calculus aids in transforming physical principles into usable formulas.

• Differentiation revealed the tangential nature of velocity: \(\vec{v}= A \omega \cos \omega t \hat{\imath} - A \omega \sin \omega t \hat{\jmath}\).
• Following this, derivation also facilitated finding constant speed \(A\omega\) and angular speed \(\omega_{angular} = \omega\).

These mathematical steps are the backbone of understanding these concepts, providing clarity and precision in modeling physical movements. Consistently, mathematical derivation underpins explanations and proofs that are pivotal in physics.